I will briefly recall the notion of stable surfaces and of the corresponding moduli space. Then I will outline a partial description of the boundary points in the case of surfaces with $K^2=1$, $p_g=2$ (joint work with Stephen Coughlan, Marco Franciosi, Julie Rana and Soenke Rollenske, in various combinations) and, time permitting, in the case of Campedelli and Burniat surfaces (joint work with Valery Alexeev).

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One approach toward dealing with intractable graph problems is structural parameterization which is defined as investigation of computational complexity of NP-hard graph problems measured as a function of the structural properties of the input graph. The structure of a graph can be measured using some well-known and well-studied parameters such as treewidth, tree-depth, clique-width, vertex cover and neighborhood diversity. The main question, here, is that, for an NP-hard problem, what is the algorithmic cost of generalizing a structural parameter of the input. In this talk, after introducing some of these parameters, we give a short (and not thorough) survey on the tools such as DP and ILP techniques using in investigation of the complexity of hard problems whose input space is confined on a class of graphs with a specified structure (e.g. bounded treewidth graphs). Moreover, we give a brief introduction to the theory of parameterized complexity and classifying NP-hard problems into classes such as FPT and W[1]-hard in terms of their tractability with respect to structural parameters of the input graph.

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Meeting ID : 896 0036 6073
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Kostochka and Woodall (2001) conjectured that the square of every graph has the same chromatic number and list chromatic number. In 2015 Kim and Park disproved this conjecture for non-bipartite and bipartite graphs. It was asked by several authors whether this conjecture holds for bipartite graphs with small degrees, claw-free graphs, or line graphs. In this talk, we introduce cubic counterexamples to this conjecture to solve three open problems posed by Kim and Park~(2015), Kim, Kwon, and Park~(2015), and Dai, Wang, Yang, and Yu~(2018). In particular, we disprove a planar version of this conjecture proposed by Havet, Heuvel, McDiarmid, and Reed (2017).
Moreover, we present a revised version for this conjecture in bipartite graphs $G$ by imposing the condition $chi(G^2) ge frac{1}{2} Delta(G^2)+1$, where $G^2$ denotes the square of $G$ and the other two parameter denote its chromatic number and maximum degree. This new version still implies the List Total Coloring Conjecture. On the other hand, we introduce a family of counterexamples obtained from line graphs $L(G)$ of bipartite planar graphs $G$ with sufficiently large maximum degree satisfying $chi(L(G)^2) ge frac{1}{2} Delta(L(G)^2)+n‎$‎, where $n$ is a sufficiently large integer.
Finally, we present non-chromatic-choosable graphs with bounded maximum degree in bipartite and planar graphs. Consequently, we improve some results due to Mirzakhani (1996), Bessy, Havet, and Palaysi (2002), and Glebov, Kostochka, and Tashkinov (2005). In addition, we characterize non-chromatic-choosable graphs of order at most $9$ and settle a question posed by Nelsen~(2019).

https://us06web.zoom.us/j/89600366073?pwd=RMHt0eGdvGtkaDBG1Me3y8bOLgooGh.1
Meeting ID : 896 0036 6073
Passcode : 290109